Graphing Derivatives

Question 1

What is the notation for the first derivative of f(x)?

Answer 1

f’(x)

Question 2

What is the notation for the second derivative of f(x) and the first derivative of f’(x)?

Answer 2

f’‘(x)

Question 3

What is an inflection point?

Answer 3

A point where the original function f(x) changes concavity; f’‘(x)=0 at an inflection point on f(x).

Question 4

What is a critical point?

Answer 4

A point on the derivative where f’(x)=0 or f’(x)=DNE.

Question 5

If the first derivative has an x-intercept going from positive to negative and the second derivative is less than zero…

Answer 5

…there is a maximum at that point on f(x).

Question 5

If the first derivative has an x-intercept going from negative to positive and the second derivative is greater than zero…

Answer 5

…there is a minimum at that point on f(x).

Question 6

If the first derivative has a point of local extrema (maximum, minimum, or shelf) and the second derivative equals zero…

Answer 6

…there is an inflection point at that point on f(x).

Question 7

When f’‘(x) is positive, what is f(x)?

Answer 7

concave up

Question 8

When f’‘(x) is negative, what is f(x)?

Answer 8

concave down

Question 9

When f(x) is increasing, f’(x) is…

Answer 9

positive

Question 10

When f(x) is decreasing, f’(x) is…

Answer 10

negative

Question 11

Features of Graph Order

Answer 11

concave up/concave down > increasing/decreasing >
positive/negative

Question 12

What are the derivatives of sin(x) and cos(x)?

Answer 12

sin(x)=cos(x)
cos(x)=-sin(x)

Question 13

How do you order the values of points on the function, derivative, and second derivative?

Answer 13

Plug the values into the x^n of the function, derivative, or second derivative (depending on what the point value the question is asking for).